Julian Del Bel
Mathematical sciences; Pure mathematics; Algebra and number theory; Algebraic and differential geometry; Category theory, k theory, homological algebra; Lie groups, harmonic and Fourier analysis; Group theory and generalisations; Combinatorics and discrete mathematics (excl. physical combinatorics); Operator algebras and functional analysis; Mathematical logic, set theory, lattices and universal algebra; Ordinary differential equations, difference equations and dynamical systems; Partial differential equations; Pure mathematics not elsewhere classified; Real and complex functions (incl. several variables); Topology; Applied mathematics; Applied mathematics not elsewhere classified; Approximation theory and asymptotic methods; Calculus of variations, mathematical aspects of systems theory and control theory; Complex systems; Dynamical systems in applications; Mathematical methods and special functions; Operations research; Theoretical and applied mechanics; Mathematical physics; Algebraic structures in mathematical physics; Integrable systems (classical and quantum); Mathematical aspects of classical mechanics, quantum mechanics and quantum information theory; Mathematical aspects of general relativity; Mathematical aspects of quantum and conformal field theory, quantum gravity and string theory; Mathematical physics not elsewhere classified
Julian Del Bel's Research Resources
Mathematical Formalization of Eigenvergence in Recursive Expansive Dynamics
We establish a rigorous mathematical formalization of eigenvergence, a novel convergence phenomenon in recursive expansive dynamical systems characterized by golden ratio scaling (phi^-n) and fractal self-similarity. This theoretical framework unifies concepts from spectral operator theory, algebraic deformation theory, and differential geometric topology within the context of Recursive Expansive Dynamics (REDS). The eigenvergence phenomenon emerges naturally from operators R: X → X on complete metric spaces satisfying three fundamental conditions:
(i) local expansive properties with sensitivity parameter delta > 0;
(ii) recursive structure with golden ratio spectral modulation (spectral radius rho(R) = phi^-1); and
(iii) self-similarity transformations maintaining fractal consistency.
We prove that sequences {x_n} generated by such operators converge to unique limit points x* with optimal rate |x_n - x*| ≤ C * phi^-n, establishing eigenvergence as a fundamental convergence mode distinct from classical fixed-point theory. The theoretical scaffolding encompasses four major components.
First, Gromov-Hausdorff convergence in metric space sequences with rate d_GH(X_n, X_infty) ≤ C * phi^-n.
Second, recursive Lie algebra deformations with structure constants C^k_{ij}(n) = C^k_{ij}(0) + sum_{m=1}^n phi^-m * delta C^k_{ij}(m), satisfying cohomological constraints H^2(g, g) = 0.
Third, Cykloid Influence spacetime geometries with curvature recursion R^{(n+1)}{mu nu rho sigma} = R^{(n)}{mu nu rho sigma} + phi^-n * K^{(n)}{mu nu rho sigma}, yielding fractal dimensions D_H approximately 3.48.
Fourth, eigenvergent hypergeometric functions of the form 2F1_eig(a, b; c; z) = sum{n=0}^infty [(a)_n (b)_n / (c)_n n!] * phi^-n * z^n, with convergence radius R = phi.
Physical applications generate testable predictions, including gravitational wave echoes with frequency scaling f_n = f_0 * phi^-n and cosmic microwave background angular correlations exhibiting golden ratio modulation. The mathematical formalization extends beyond classical convergence paradigms to encompass multi-scale recursive dynamics, non-local memory effects, and fractal attractor structures across algebraic, geometric, and physical domains.
https://zenodo.org/records/15724146
Mathematical Lifting of Eigenvergence in Recursive Expansive Dynamics
This lift scaffolds the core geometric, algebraic, and physical components of the refined Recursive Expansive Dynamics (REDS) framework, integrating the hierarchical structure of dimensional modulators, recursive influence pathways, and eigenvergence principles governed by the golden ratio. It consolidates foundational equations, geometric constructions, coupling mechanisms, and empirical predictions, extending the original theory with detailed mathematical formalism and physical implications.
This comprehensive analysis integrates two advanced mathematical prototypes for eigenvergence theory with the refined Recursive Expansive Dynamics (REDS) framework, creating a unified theoretical scaffolding that extends eigenvergence principles across topological, categorical, stochastic, and quantum domains
https://zenodo.org/records/15724536
The Cykloid-Adelic Recursive Expansive Field Equation (CARE)
We develop a unified field framework, the Cykloid-Adelic Recursive Expansive Field Equation (CARE), which scaffolds stratified geometric manifolds with adelic number-theoretic dynamics via recursive cycloidal parameter spaces. This approach rigorously defines a hierarchy of embedded strata, governed by golden-ratio scaling, and constructs a convergence-proof action principle on a fractal manifold. CARE introduces novel mechanisms for field trifurcation into matter, interaction, and geometric sectors; formulates curvature-based nexus point theory with discrete quantization; and derives a p-adically regulated cosmological constant. The framework delivers testable predictions in gravitational wave echoes, CMB multipole anomalies, and dark matter fractal distributions, while grounding the theory in weighted Sobolev spaces and distributional analysis on singular stratified spaces. Methods The full expression lives alongside the original paper RED34Section
1: Mathematical Framework Definitions Core Functions – r₁(t)=exp(t·α), r₂(t)=tᵅ, d(t)=2 sin(t·α)+3 Time Derivatives – ṙ₁=α exp(t α), ṙ₂=tᵅ·(α/t), ḋ=2 α cos(t α) Section
2: Partial Derivatives of the Area Function Auxiliary terms θ₁…θ₄ defined from r₁, r₂, d and Δ=√(θ₁θ₂θ₃θ₄) Exact formulas for ∂A/∂r₁, ∂A/∂r₂, ∂A/∂d, each featuring: – a Heron-type square‐root term – a cyclic product of three θ’s divided by 2 θ₁θ₂θ₃θ₄ – an arccos term – additional correction factors Section
3: Total Time Derivative of Area dA/dt = (∂A/∂r₁)ṙ₁ + (∂A/∂r₂)ṙ₂ + (∂A/∂d)ḋ Substitution of ṙ₁, ṙ₂, ḋ yields a sprawling multi-line expression combining all three partial contributions Section
4: Key Mathematical Constructs Golden ratio φ = (1+√5)/2 ≈ 1.618 Fractal Hausdorff dimension D_H ≈ 3.48 Tribonacci constant ≈ 1.839 Adelic prime-based product structures Recursive operator ℛ(x)=limₙ→∞φ⁻ⁿ P_stratumₙ∘Tⁿx Eigenvergence at rate O(φ⁻ⁿ) Section
5: Validation Metrics χ²/ν = 1.03 (ν=112) Gelman–Rubin R̂ = 1.002 ± 0.0003 Energy conservation: d(E_epic+E_epitro)/dt = 0 CMB multipoles C_ℓ ∼ ℓ^{–α} cos(2π ℓ φ) Gravitational-wave echo frequencies fₙ = f₀ φ⁻ⁿ
https://zenodo.org/records/15751227
Modified Klein-Gordon Fractal Stability and Quantum Zeno Analogies
We present a field-theoretic system through a modified Klein-Gordon framework embedded in a semi-recurve, nonlocal interaction term that governs recursive expansive dynamics across spacetime. The resulting field equation incorporates influence kernels that modulate phase coherence through fractal stability conditions and ratio scaling symmetries. We derive Von Neumann stability criteria for numerical implementations and establish a hyperfolding mechanism that enforces mode-selective suppression, analogous to the Quantum Zeno effect in constrained Hilbert spaces. We demonstrate that Hypatian-induced perturbations encode prime number geometry through hyperspherical mappings, revealing structured resonances in spectral distributions. Potential observational signatures include anomalous gravitational wave harmonics, constrained decoherence rates in quantum systems, and resonance shifts in high-energy scattering amplitudes.
https://hal.science/hal-05076360v1
Recursive Inflationary Fractal Time (RIFT) Mathematical Foundations and Physical Implications
The Recursive Inflationary Fractal Time (RIFT) framework unifies number theory, fractal geometry, and quantum gravity through the handling of divergent infinite prime products and analytic continuation. Utilizing adelic integration, zeta function regularization, and recursive entropy dynamics, RIFT attains a theoretically stable cosmological constant Λ = 1 despite inherent divergences. Numerical validations up to 60 primes confirm convergence to Λ ≈ 1 ± 10−101, with precision scaling logarithmically with prime count. Observable predictions include fractal-modulated gravitational waves and Mersenne prime-induced dark matter hierarchies. Methods The Recursive Inflationary Fractal Time (RIFT) framework addresses fundamental mathematical and physical challenges by reconciling divergent series with analytic regularization techniques. This approach unifies number theory, fractal geometry, and quantum gravity through a novel treatment of infinite prime products and their analytic continuation. The formulation incorporates: • Adelic integration balancing real and p-adic number fields, establishing a duality principle that resolves the indeterminate ∞ · 0 expressions arising from divergent-vanishing product pairs. • Zeta function regularization resolving infinite prime products through analytic continuation to s = 1 2 , where ζ(1/2) ≈ −1.4603545 provides a finite value for otherwise divergent Euler products. • Recursive entropy dynamics with golden ratio (φ) scaling, ensuring physical consistency through self-similar information growth patterns that mirror inflationary cosmological expansion. • Hyperspherical prime embedding in 4D spacetime, generating fractal lattices with angular separations scaled by φ = 1+√ 5 2 , establishing a geometric foundation for prime-modulated physical phenomena. These elements collectively impose stability on the cosmological constant (Λ = 1 ± 10−101 for 60 primes) and predict novel fractal structures in cosmology, including φ-modulated gravitational wave echoes and Mersenne prime-derived dark matter hierarchies. Numerical validations demonstrate precision scaling logarithmically with prime count ( ∼ exp(−cN/ ln N)), achieving "14.07 orders" of refinement beyond initial calculations.
https://zenodo.org/records/15106886
Rigorous Formalization of the Cykloid Strata
This introduction establishes the cycloid's importance in both mathematical theory and its practical implications in physics, setting a strong foundation for exploring more advanced topics like "Cykloid Strata." By varying the radius r and the position of the tracing point d, we have developed a framework for defining "Cykloid Strata" as distinct layers within a parameter space. This approach allows us to explore how different parameters influence the resulting cycloidal structures and provides a means of classifying them according to their geometric properties. The use of scaling laws and comparisons between different types of cycloids and trochoids enriches our understanding of the multi-layered nature of this theoretical framework, offering potential applications in both mathematics and physics.
https://zenodo.org/records/15069479
- Recursive Fractal Spacetime
- Recursive Fractal Spacetime Dynamics
- Fractal Geometry and Observational Cosmology
- Recursive Inflationary Fractal Time
- Adelic Mathematics and p-adic Structures
- Recursive Adelic Dynamics
- Adelic Aligned p-adic Spacetime
- P-Adic Structures in Energy-Momentum Tensor
- Cykloid Framework
- The DelBelian Cykloid Framework
- Cykloid-Adelic Recursive Field Equation (CARFE)
- Enhanced Cykloid Strata Framework
- Quantum Gravity and Holography
- Fractal Quantum Gravity Visualization
- Quantum Zeno Dynamics and Fractal Holography
- Recursive Fractal and Nonlocal Structures in Holography
- Number Theory and Mathematical Foundations
- Diophantus On Polygonal Numbers
- Prime-Modulated Geometry
- The Prime Cutoff at 282,281
Key Concepts
- Recursive Expansive Dynamics (RED)
- Hypatian Mathematics
- Fractal Adelic Scaffolding
- Triplexor-Adelic Hyperfolding Strata
- Cykloid-Adelic Recursive Field Equation (CARFE)
Additional profile links
Publications
- A Comprehensive Theoretical Framework for Spatiotemporal Influence in Gravitational Wave Propagation and Spacetime Geometry
- Recursive Expansive Dynamics
- Recursive Spacetime, Fractal Entropy, and Quantum Geometry
- Quantum-Consistent Adelic Integration
- Quantum-Consistent Adelic Integration and Structure of Egyptian Fractions
- Dirichlet Eta Function of Prime Distributions, Fractal Geometry, and Gravitational Wave Dynamics
- Recursive Fractal Spacetime Dynamics in Renormalization Group Flow and Higher-Spin Holography
- Hypatian Axioms Foundations of Geometric Physics
- Fractal Geometry, Recursive Dynamics, and Observational Cosmology
- Recursive Inflationary Fractal Time (RIFT) Mathematical Foundations and Physical Implications
- Adelic Aligned p-adic Spacetime
- Hypatian Power Law
- Modified Klein-Gordon Fractal Stability and Quantum Zeno Analogies
- Hypatian Math
- Recursive Expansive Dynamics in Spacetime Quantum Nonlocality and Higher-Dimensional Dynamics
- Diophantus' On Polygonal Numbers via Recursive Fractal Dynamics and Prime-Modulated Geometry
- Dimensional Energy Scaling and Fractal Geometry in Adelic Aligned p-adic Spacetime
- Fractal-Torsional-Spectral Extensions to Wiles' Modularity Theorem
- Rigorous Formalization of the Cykloid Strata
- Recursive Adelic Dynamics
- Recursive-Dimensional Mathematics From Prime Fractals to Hyperspace Geometry
- Hypatia's Math
- Spatiotemporal ϕ/π Scaling: A Framework for Gravitational, Quantum, and Causal Influence
- Euclidean Framework for Recursive Feedback Dynamics and Multifractal Analysis
- Influentia Overversum
- Recursive Expansive Dynamics in Spacetime (REDS) Quantum Nonlocality and Higher-Dimensional Dynamics
- Recursive Expansive Hypergeometric Calculus
- Diophantus' On Polygonal Numbers via Recursive Fractal Dynamics and Prime-Modulated Geometry
- Recursive Adelic Dynamics, Fractal Holography, and Quantum Geometry for Spacetime, Gravity, and Quantum Information
- The E8 to G2 Symmetry Breaking Mechanism of Adelic Curvature and Exceptional Lie Scaffolding: Euclid's Fifth Postulate as a Quantum Gravitational Phase Indicator
- Categorical Reformation of Arithmetic and Spacetime Geometry Through Fractal-Adelic Scaffolding
- Recursive Expansive Dimensionless Framework for Gravitational Echoes and Fractal Spacetime
- The E8 to G2 Symmetry Breaking Mechanism of Adelic Curvature and Exceptional Lie Scaffolding: Euclid's Fifth Postulate as a Quantum Gravitational Phase Indicator
- Recursive-Adelic-Fractal Scaffolding and Its Connection to Monstrous Moonshine
- Inverse Zero Operator (IZO) Idempotence
- Mathematical Lifting of Eigenvergence in Recursive Expansive Dynamics
- Addendum to ``The E8 to G2 Symmetry Breaking Mechanism of Adelic Curvature and Exceptional Lie Scaffolding: Euclid's Fifth Postulate as a Quantum Gravitational Phase Indicator''
- Adelic Fractal Cosmogenesis Recursive Number-Theoretic Scaffolding of Spacetime Geometry and Cosmic Observables
- Proof of the Adelic Weil Conjectures for the Zeta Function ζMBer(s)
- Mathematical Formalization of Eigenvergence in Recursive Expansive Dynamics
- Hypatian Physics Lean4-Verified Framework Recursive Expansive Algebraic-Geometric Quantum Gravity, Particle Physics, and Cosmology
- Recursive-Expansive Dynamics Frameworks Formalization with Negation Isolation and Inverse Zero Operators
- Mind Blind Aphantasia, Thought Process Devoid of Visualisations
- JulianDelBel/Adelic: Escherian Mathematical Mechanisms in Recursive-Expansive Dynamics
- JulianDelBel/Adelic: Recursive-Expansive Dynamics
- The Cykloid-Adelic Recursive Expansive Field Equation (CARE)
- Adelic Aligned p-adic Spacetime and Recursive Expansive Dynamics
- Cyclotomic Fields as Recursive Time Structures